The mathematical theory of games in the form known to economists, social scientists and biologists was developed by John von Neuman and Oskar Morgenstern. Their framework has been deepened and generalized over the past seven decades. Psychologists use classical probability theory in which rational choice is built in, and they interpret any departure from classical probabilities as irrationality. It has been suggested that an alternative interpretation would be to retain rationality by giving up classical probability theory in favour of quantum probability theory. Some game theorists have demonstrated that a tit-for-tat strategy evolves in the repeated Prisoners’ Dilemma setting where one player responds back in the same coin to the other after seeing the letter’s move. Co-operation is generated at times since ‘credible’ threat can be created in repeated games. The main difference between quantum and classical probability theories is the use of Hilbert space representations of states rather than set theory.
The mathematical theory of games in the form known to economists, social scientists and biologists was developed by John von Neuman and Oskar Morgenstern. Their framework has been deepened and generalized over the past seven decades. Psychologists use classical probability theory in which rational choice is built in, and they interpret any departure from classical probabilities as irrationality. It has been suggested that an alternative interpretation would be to retain rationality by giving up classical probability theory in favour of quantum probability theory. Some game theorists have demonstrated that a tit-for-tat strategy evolves in the repeated Prisoners’ Dilemma setting where one player responds back in the same coin to the other after seeing the letter’s move. Co-operation is generated at times since ‘credible’ threat can be created in repeated games. The main difference between quantum and classical probability theories is the use of Hilbert space representations of states rather than set theory.
In cognitive psychology, some experiments of games were reported [1, 2, 3, 4], and these demonstrated that real players did not use the "rational strategy" provided by classical game theory. To discuss probabilities of such "irrational choice", recently, we proposed a decision-making model which is based on the formalism of quantum mechanics [5, 6, 7, 8]. In this paper, we briefly explain the above model and calculate the probability of irrational choice in several prisoner's dilemma (PD) games.
Of the three faculties of subjectivity, the empirical case for quantum cognition is the strongest. In the past decade a rapidly growing body of scholarship has emerged at the intersection of mathematical psychology and physics that not only models cognition in explicitly quantum terms, but has rigorously tested these models as well. While it remains to be seen how mainstream psychologists will respond, “quantum decision theory” has clearly come of age. Moreover, at the same time an almost completely independent literature has emerged on quantum game theory as well. This work is more in a purely formal vein and thus less well tested against real data, although its predictions about how people should behave in quantum games appear to conform to what experimental game theorists have found. I briefly take up quantum game theory at the end of this chapter, but my main focus will be on its decision-theoretic cousin. Quantum consciousness theory suggests that human beings are literally walking wave functions. Most quantum decision theorists would not go that far, and indeed – perhaps wary of controversy – they generally barely mention quantum consciousness, and then only to emphasize that they are making no claims about what is going on deep inside the brain (much less about consciousness), but are only interested in behavior. Instead, they have motivated their work in two other ways. First, they have highlighted the intuitive fit between quantum theory's indeterminism and the probabilistic character of human behavior. If we were founding the social sciences today and started – without metaphysical prejudice – by looking for useful models in physics, then quantum models would be a much more obvious import than classical ones. Second, quantum decision theorists have argued that their model can explain long-standing anomalies in human behavior under uncertainty. Beyond that they have remained agnostic about quantum consciousness, which makes quantum decision theory a form of what is called “weak” or “generalized” quantum theory, in which the formalism is detached from its physical basis and used simply as a tool to describe phenomena.
Recent results from cognitive psychology suggest that human behavior can be, in some situations, better described by laws known from physical quantum theory. This work responds to this development and seeks to answer the question of how quantum models of cognition and decision-making can contribute to our understanding of the social world. We introduce the approaches to the application of quantum theory in social sciences - quantum consciousness, quantum reasoning and quantum metaphor and further develop the second, in the context of strategic interaction in game theory. We show that the quantum model corresponds to both the psychological intuition and many empirically known anomalies such as the order effect, disjunction effect, or context effect. In this work, we derive a specific model of strategic interaction between two players, and we test it in the experimental Prisoner's Dilemma game. The model is based on the rational choice theory, which is nevertheless re-defined when it is shown how the strategic choice is changed if the quantum probability (C*-algebra) is used as the underlying theory instead of classical probability (σ-algebra). The results obtained support the quantum model and showed a different representation of the game among various groups of players. The quantum model of...
Two experimental tasks in psychology, the two-stage gambling game and the Prisoner's Dilemma game, show that people violate the sure thing principle of decision theory. These paradoxical findings have resisted explanation by classical decision theory for over a decade. A quantum probability model, based on a Hilbert space representation and Schrödinger's equation, provides a simple and elegant explanation for this behaviour. The quantum model is compared with an equivalent Markov model and it is shown that the latter is unable to account for violations of the sure thing principle. Accordingly, it is argued that quantum probability provides a better framework for modelling human decision-making.
A small-but-growing number of social scientists have in recent years begun to explore the purchase of formalisms and a probability theory originally developed to accommodate nonclassical experimental results in particle physics. Motivated by the desire to explain empirical outcomes that do not fit comfortably with the axioms of rational choice theory—and that are thus related tenets of classical probability theory—these scholars have begun to examine and embrace the theory of probability namesaked for Max Born as a general framework for understanding and modeling choice. Born’s (1926) account of probability differs from the familiar, classical context of Kolmogorov both in terms of its mathematical exposition/foundation and with respect to its governing axioms. The Born theory is expressly geometric as opposed to set-theoretic. Its axioms are formalized in the (usually) complex planes of Hilbert spaces, where distances are most generally conceptualized as metrics between (sometimes high-dimensional) spaces as opposed to points. In this framework, certain of Kolmogorov’s set-theoretic axioms can be relaxed, and empirical results that do not agree with them can be systematically accommodated. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The present thesis (mostly) concerns the application of alternative mathematical methods to understand patterns in human cognition and to model them. The different chapters presented in this thesis show research that concerns the application of quantum probability (QP) theory in the modeling of human decision-making. Quantum probability (QP) theory is a theory for how to assign probabilities to events. QP theory can be thought of as the probability rules from quantum mechanics, without any of the physics. This work is not about the application of quantum physics to brain physiology. Rather, we are interested in QP theory as a mathematical framework for cognitive modelling. This theory is potentially relevant in any behavioural situation that involves uncertainty. QP theory is analogous to classical probability theory, though QP theory and classical probability (CP) theory are founded from different sets of axioms (the Kolmogorov and Dirac/von Neumann axioms respectively) and so are subject to alternative constraints. In this thesis we show that especially over the last decade, there has been a growing interest in decision-making and cognitive models using a quantum probabilistic (QP) framework. We see how this development encompasses publications in major journals (see Pothos and Busemeyer, 2013; Wang et al., 2014; and Yearsley and Pothos, 2014; among others), special issues, and dedicated workshops, as well as several comprehensive books (Busemeyer and Bruza, 2012; Khrennikov, 2010; and Haven and Khrennikov, 2010). However, uncertainty itself is neither ethical nor unethical – yet it is inherent to most situations in which, for instance, moral judgments and decisions have to be made. For a descriptive understanding of judgment and decisions in moral situations, it is an important lesson to acknowledge both the cognitive side (bounded rationality) and the environment (ecological rationality) – and thus the uncertainty of the world and how the mind deals with it. This thesis also shows significant interest in moral and social psychology. Specifically, we consider present technologies that suggest a need for evaluating alternative contexts for ethical decision-making. How the research on human-machine interaction feeds back into humans’ understanding of themselves as moral agents? This key question ultimately relates to the nature of ethical theory itself. Overall, this dissertation presents and addresses not only standard aspects of decision- making processes, such as similarity judgments (Chapters 1 to 4) or the constructive role of articulating impressions (Chapter 5), but also standard aspects of social psychology, such as moral judgments (Chapters 6 and 7) and game theory (Chapter 8). As stated in the Declarations section, the present thesis is a combination of a standard and a publication- based dissertation.